# On the dimension of downsets of integer partitions and compositions

## TL;DR

This paper characterizes when downsets of integer partitions and compositions have finite dimension, showing all proper downsets of partitions are finite-dimensional and identifying minimal infinite-dimensional downsets for compositions.

## Contribution

It provides a complete characterization of finite and infinite-dimensional downsets in the posets of partitions and compositions, including minimal infinite-dimensional examples.

## Key findings

- All proper downsets of partitions have finite dimension.
- Four minimal infinite-dimensional downsets of compositions are identified.
- Downsets not containing these four are finite-dimensional.

## Abstract

We characterize the downsets of integer partitions (ordered by containment of Ferrers diagrams) and compositions (ordered by the generalized subword order) which have finite dimension in the sense of Dushnik and Miller. In the case of partitions, while the set of all partitions has infinite dimension, we show that every proper downset of partitions has finite dimension. For compositions we identify four minimal downsets of infinite dimension and establish that every downset which does not contain one of these four has finite dimension.